The main condition necessary for the advancement of physics and astronomy that progressed during the Scientific Revolution was the advance of mathematics, which allowed the proof of abstract theories and provided a more logical method for attacking the Aristotelian system.
After the decline of Greek mathematics, and a period in which religious authority stultified intellectual creativity, the renaissance opened a new era which leads to the revitalization of science and mathematics. Descartes' philosophical methodology, involving scepticism (rejection of authority) and rationalism (an assertion of self-confidence) shows the shift of mood. Though Descartes' method is deductive, he rejects (Aristotelian) logic as sterile.
Gallileo provided a new view of science, emphasising descriptive and quantitative explanation. Science for Galileo involves the derivation of a body of knowledge from a small number of mathematically formulated physical principles. This new science, most clearly illustrated in Newton's mechanics, provided the stimulus for major developments in mathematics in the 17th century and the rapid development of analysis through the 18th.
Co-ordinate Geometry Co-ordinate geometry was introduced by Fermat and Descartes, ignoring rather than solving the foundational problems which had prevented the Greeks from taking this step (viz: the lack of any well-understood number system which could account for incommensurable ratios). This development is important to science because it makes geometry quantitative and permits the use of algebraic methods. Geometry must be quantitative for it to be useful in science and engineering, and algebraic methods permit the more rapid development of mathematics than the less systematic (if more rigorous) methods required by the Greek axiomatic approach to geometry.
The Calculus Developed independently by Newton and Leibniz, the calculus is the foundation on which a large part of the mathematics required for science is built. The development of the calculus depended upon a number system which not only includes irrational (numbers necessitated by incommensurable ratios), but even infinitesimal numbers, which sometimes behave like zero and sometimes don't. Objections, particularly to the latter, failed to halt the onward sweep of mathematics in the service of science.
Analysis During the 18th century, the calculus was further developed and other branches of analysis were rapidly opened up. These developments lacked logical rigour and were guided by intuitive and physical insights. Though the lack of rigour was noted, for example by Bishop Berkeley, and many attempts were made to put things right, this did not inhibit the development of the subject. It was not until well into the 19th century that rigorous foundations for this plethora of new mathematics could be discovered.
The Formalisation of Mathematics
During the period from about 1821 to 1908, which begins with the publication of Cauchy's Coursd'analysealgebraic and concludes with the publication in 1908 of Russell's Mathematical Logic as Based on The Theory of Types and Zermelo's first order axiomatisation of set theory, mathematicians restored and surpassed the standards of rigour which had been established during the period of classical greek mathematics but neglected during the mathematization of science.
The Rigourisation of Analysis The use of infinitesimals was one of the most conspicuous practices in the development of analysis for which no justification or rationale could be offered. The rigourisation of analysis begins with the elimination of infinitesimals in favour of arguments using limits. Frege's Logic For millenia, mathematics had been a science based on deductive logic. But no account of logic had ever been produced which was adeqate for the purposes of mathematics. By abandoning Aristotle's subject-predicate analysis of the form of sentences, Frege was able to devise a logic which would prove sufficient for the formalisation of mathematics. Grundgesetze and Principia Mathematica: Frege's logic was intended to provide a foundation for mathematics, and Frege showed how this could be done in his Grundgesetze der Arithmetic. This however proved to be based on an inconsistent logical system. Russell, taking the paradoxes into account, devised his Theory of Types and demonstrated with Whitehead in Principia Mathematicahow this could be used to formalise mathematics. Formal mathematics was shown to be rather hard work for human beings, and did not catch on.
The Theory of Real Numbers The rigourisation of analysis preceded and showed the need for a better account of the real number system, since the necessary proofs of results in analysis depended on properties of real numbers which were not themselves rigorously established.
Cantorian Set Theory Both the theory of real numbers and the idea of a function depended upon an informal notion of set. Cantor turned the very simple idea of a set into a rich theory which was to become the foundation of modern mathematics.
Zermelo's Axiomatic Set Theory After the paradoxes became conspicuous in set theory it became essential to have a consistent axiomatization of the theory if it was to be used. Zermelo's axiomatisation was the first, and with some subsequent improvements became the de-facto standard foundation for mathematics. Though Zermelo's account was informal it is a small step from there for a fully formal account of mathematics as based on first order set theory.
What changed in mathematics
LETTERS TO REPRESENT UNKNOWN QUANTITIES. During the late sixteenth century, a French lawyer, Francois Viete, was among the first to use letters to represent unknown quantities. In 1591 and after, he applied this algebraic method to geometry, laying the foundation for the invention of trigonometry. THE INCLINE PLANES- FRACTIONS. The Fleming Simon Stevin also worked with geometry during the late sixteenth century, applying it to the physics of incline planes and the hydrostatic surface tension of water. Additionally, he introduced the decimal system of representing fractions, an advance which greatly eased the task of calculation. INVENTION OF LOGARITHMS. However, perhaps the most important mathematical advance of the early period of the Scientific Revolution was the invention of logarithms in 1594 by John Napier of Scotland. Napier spent the next 20 years of his life developing his theory and computing an extensive table of logarithms to aid in calculation. In 1614, he published Description of the Marvelous Canon of Logarithms, which contained the fruits of these labors. TRAJECTORIES OF MOTION AND CARTESIAN CO-ORDINATES. Johannes Kepler also did a great deal of work in geometry, which proved significant to his subsequent work in astronomy. In 1637, Rene Descartes publishedGeometry,in which he describes how geometry relates to motion and showed that at any moment the position of a point can be defined by its relation to surrounding planes or reference. The most well-known application of this theory is the use of a curve on a graph to represent the motion of an object, which could then be defined by a mathematical equation.. DIFFERENTIAL CALCULUS. Further progress in mathematics was made by Oxford professor John Wallis. His first work,Arithmetica Infinitum,published in 1655, set the stage for the invention and development of differential calculus. Wallis' became one of Isaac Newton's major influences. Wallis was the first mathematician to apply mathematics to the operation of the tides, and also invented the symbol used to denote infinity. FUTHERMORE. Many mathematicians applied their knowledge to the study of optics, a field that had garnered great interest since the Middle Ages. The advances made in this field, including the development of techniques for higher resolution, led to the better construction of optical instruments, such as the telescope, which played a large part in the later work of Galileo.
Summing up 1.It was a "gradual" revolution
The Scientific Revolution was not a revolution in the sense of a sudden eruption ushering in radical change, but a century-long process of discovery in which scientists built on the findings of those who had come before — from the scientific achievements of the ancient Greeks to the scholarly contributions of Islamic thinkers, to the work of certain late-medieval and early-Renaissance Europeans. The expanding economy of the Age of Discovery represented another significant impulse, in that the need for better navigation, time-keeping, and naval engineering pushed Europeans to pose new questions and, in turn, devise new methods to solve them.
2. Why in Europe ?
Modern European history in the perspective of world history
Europe: cultural innovations and navigation around the world.
Asia (China and Japan): 1600s: a closed or semi-closed system to maintain existent social hierarchy; where government office was worshipped by the people and technology and commerce were treated lightly by the state.
3.The pre-Scientific Revolution thinking
The law of hypotheses: human limitations of knowledge led to the human inability to know the whole truth.
Hypotheses were the best humans could achieve in understanding the world. But what Galileo and others tried to demonstrate was that human observations conveyed truth.
4. Stages of scientific development
Based on theology and Aristotle.
Based on Hermetic theory: a divine spirit present in all the material things in the world. And the job of the natural philosopher was to capture these divine messages.
A completely secular approach to science, treating the world as consisting of a rational, knowable order.
5. Why was mathematics so important to the scientists of the 16th and early 17th centuries?
The truth of God could be found in math.
Implied in the Hermetic theory is an independent search for truth instead of accepting established truth; mathematics enabled independent thinking beyond a mere reading of the scriptures.
The mathematical conclusion can be publicly arrived at and publicly demonstrated, encouraging social consensus.
6. What were the main contributions of Copernicus, Brahe, Kepler, Galileo, and Newton to new scientific ways of thinking of the universe?
Copernicus: changed the view of the universe from geo-centric to helio-centric. Earth moved in perfect circles around the sun. Earth’s self-rotation.
Brahe: in Denmark, Changed established the view of the stars. Royal patronage.
Kepler: Three laws of planetary motion: challenging the view of a harmonious universe through the perfect circles of the stars’ movements, and through the constant speed of these movements.
Galileo: the telescope to see the moon; inertia: contradicts the traditional belief that objects are naturally at rest; the state of motion is just as natural as the state of rest.
Newton: Universal law of gravity; calculus; nature of light; mathematical rules of the three laws of motion: inertia, acceleration, action and reaction; the universe was infinite and had no center.
9. TheScientific methods
The development of scientific discoveries was justified and facilitated by scientific philosophy or methods. Newton tried that. But more than anyone else, it was Rene Descartes who created a justification connecting mathematics with physics, and individual observations with religion.
10. The importance of Descartes
René Descartes (1596–1650) is considered one of the major figures of the seventeenth century and an important contributor to the Scientific Revolution. Descartes was a French-born philosopher and mathematician but spent most of his adult life living in the Dutch Republic. The Dutch environment influenced the development of Descartes’s philosophical and mathematical theories. Although Descartes is primarily remembered for his writings as a philosopher—today he is often considered the father of modern philosophy—his findings in mathematics were, perhaps, equally innovative and influential. Published in 1637, La Géometr, was Descartes’s major treatise in mathematics. In it he described the principles of coordinate geometry, a system he created for studying geometry using algebraic principles and a coordinate system. Also known as “analytic” or “Cartesian” geometry, this system allowed geometric problems to be solved more easily through the use of algebra. Descartes’s creation provided a link between algebra and geometry. Cartesian geometry presented algebraic equations in the form of geometric shapes mapped onto a coordinate system. As a result of this creation, Descartes is referred to as “the father of analytical geometry.” Descartes’s mathematical writings and findings laid the groundwork for future discoveries in mathematics, such as infinitesimal calculus.
11.What are the four main principles of his method?
To accept nothing as true unless it was proved.
To divide each difficulty into many parts and search for a solution.
To arrange thoughts in order beginning with the simplest to the most difficult.
To make sure nothing is omitted.
12.How is Descartes’s philosophy affected by the ideas of new science?
The idea of proof or testing was new, and could only exist when the truth was knowable to human beings. A collection of data; and procedure of proof (from the simple to the difficult) made proof a publicly demonstrable procedure.
Clear and distinct ideas: emphasis on exactness and accuracy.
Science also emphasizes exhaustion of data, so that the results can apply to similar conditions every where in the universe.
Emphasis on procedure suggests that science had its own internal rules to follow.
Take apart a big problem and make it observable.
13.How did Descartes justify observation?
Begin with a clear and distinct idea (based on math), then you seek it in reality because it will sure be there. In other words, explorations in the world are valid because they are backed by these "perfect ideas" of God in one’s mind. This can be used to justify empirical observations only because it made observations ‘legitimate,’ so to speak, by reconciling Catholicism and scientific observation, not because his observation was the same as our definition of observation today.
During the period from about 1821 to 1908, which begins with the publication of Cauchy's Cours d'analyse algebraic and concludes with the publication in 1908 of Russell's Mathematical Logic as Based on The Theory of Types and Zermelo's first order axiomatisation of set theory, mathematicians restored and surpassed the standards of rigour which had been established during the period of classical greek mathematicsbut neglected during the mathematisation of science
The term "Scientific Revolution" is a modern one. Most Early Modern scholars called themselves "natural philosophers" rather than "scientists." Both institutionally and conceptually, science was not the independent practice it is today. Much of what we know as science originally belonged to the study of philosophy and theology, and most often was carried out under Church or court patronage.
1.A "gradual" revolutioin
The Scientific Revolution was not a revolution in the sense of a sudden eruption ushering in radical change, but a century-long process of discovery in which scientists built on the findings of those who had come before — from the scientific achievements of the ancient Greeks to the scholarly contributions of Islamic thinkers, to the work of certain late-medieval and early-Renaissance Europeans. The expanding economy of the Age of Discovery represented another significant impulse, in that the need for better navigation, time-keeping, and naval engineering pushed Europeans to pose new questions and, in turn, devise new methods to solve them.
2. Why Europe?
Modern European history in the perspective of world history
Europe: cultural innovations and navigation around the world.
Asia (China and Japan): 1600s: a closed or semi-closed system to maintain existent social hierarchy; where government office was worshipped by the people and technology and commerce were treated lightly by the state.
3.Pre-Scientific Revolution thinking
The law of hypotheses: human limitations of knowledge led to human inability to know the whole truth. Hypotheses were the best humans could achieve in understanding the world. But what Galileo and others tried to demonstrate was that human observations conveyed truth.
4. Stages of scientific development
Based on theology and Aristotle.
Based on Hermetic theory: a divine spirit present in all the material things in the world. And the job of the natural philosopher was to capture these divine messages.
A completely secular approach to science, treating the world as consisting of a rational, knowable order.
5. Why was mathematics so important to the scientists of the 16th and early 17th centuries?
The truth of God could be found in math.
Implied in the Hermetic theory is independent search for truth instead of accepting established truth; mathematics enabled independent thinking beyond a mere reading of the scriptures.
Mathematical conclusion can be publicly arrived at and publicly demonstrated, encouraging social consensus.
6. What were the main contributions of Copernicus, Brahe, Kepler, Galileo, and Newton to new scientific ways of thinking of the universe?
Copernicus: changed the view of the universe from geo-centric to helio-centric. Earth moved in perfect circles around sun. Earth’s self-rotation.
Brahe: in Denmark,. Changed established view of the stars. Royal patronage.
Kepler: Three laws of planetary motion: challenging the view of a harmonious universe through the perfect circles of the stars’ movements, and through the constant speed of these movements.
Galileo: telescope to see the moon; inertia: contradicts the traditional belief that objects are naturally at rest; the state of motion is just as natural as the state of rest.
Newton: Universal law of gravity; calculus; nature of light; mathematical rules of the three laws of motion: inertia, acceleration, action and reaction; the universe was infinite and had no center.
9. Scientific methods
The development of scientific discoveries were justified and facilitated by scientific philosophy or methods. Newton tried that. But more than any one else, it was Rene Descartes who created a justification connecting mathematics with physics, and individual observations with religion.
10. Descartes
11. What are the four main principles of his method
To accept nothing as true unless it was proved.
To divide each difficulty into many parts and search for solution.
To arrange thoughts in order beginning with the simplest to the most difficult.
To make sure nothing is omitted.
12. How is Descartes’s philosophy affected by the ideas of new science?
Idea of proof or testing was new; and could only exist when truth was knowable to human beings.
Meticulous collection of data; and procedure of proof (from the simple to the difficult) made proof a publicly demonstrable procedure.
Clear and distinct ideas: emphasis on exactness and accuracy.
Science also emphasizes exhaustion of data, so that the results can apply to similar conditions every where in the universe.
Emphasis on procedure suggests that science had its own internal rules to follow.
Take apart a big problem and make it observable.
13. Definition of absolute certainty
Mathematical formulae.
He bridged physics and mathematics: by reducing the whole world to mathematical representation. He also bridged observation and legitimacy by connecting observation through mathematics. Furthermore, he justified truth through mathematics and observation by connection with God.
14. How did Descartes justify observation?
Begin with a clear and distinct idea (based on math)
Then you seek it in reality because it will sure be there.
In other words, explorations in the world are valid because they are backed by these "perfect ideas" of God in one’s mind. This can be used to justify empirical observations only because it made observations ‘legitimate,’ so to speak, by reconciling Catholicism and scientific observation, not because his observation was the same as our definition of observation today.
||The Rigourisation of Analysis The use of infinitesimals was one of the more conspicuous practices in the development of analysis for which no justification or rationale could be offered. The rigourisation of analysis begins with the elimination of infinitesimals in favour of arguments using limits.
Frege's Logic For millenia mathematics had been a science based on deductive logic. But no account of logic had ever been produced which was adeqate for the purposes of mathematics. By abandoning Aristotle's subject-predicate analysis of the form of sentences, Frege was able to devise a logic which would prove sufficient for the formalisation of mathematics.
Grundgesetze and Principia Mathematica Frege's logic was intended to provide a foundation for mathematics, and Frege showed how this could be done in his //Grundgesetze der Arithmetic//. This however proved to be based on an inconsistent logical system. Russell, taking the paradoxes into account, devised his Theory of Types and demonstrated with Whitehead in //Principia Mathematica// how this could be used to formalise mathematics. Formal mathematics was shown to be rather hard work for human beings, and did not catch on.
**The Theory of Real Numbers** The rigourisation of analysis preceded and showed the need for a better account of the real number system, since the necessary proofs of results in analysis depended on properties of real numbers which were not themselves rigorously established.
Cantorian Set Theory Both the theory of real numbers and the idea of a function depended upon an informal notion of set. Cantor turned the very simple idea of a set into a rich theory which was to become the foundation of modern mathematics.
Zermelo's Axiomatic Set Theory After the paradoxes became conspicuous in set theory it became essential to have a consistent axiomatisation of the theory if it was to be used. Zermelo's axiomatisation was the first, and with some subsequent improvements became the de-facto standard foundation for mathematics. Though Zermelo's account was informal it is a small step from there for a fully formal account of mathematics as based on first order set theory.
René Descartes (1596–1650) is considered one of the major figures of the seventeenth century and an important contributor to the Scientific Revolution. Descartes was a French-born philosopher and mathematician, but spent most of his adult life living in the Dutch Republic. The Dutch environment influenced the development of Descartes’s philosophical and mathematical theories. Although Descartes is primarily remembered for his writings as a philosopher—today he is often considered the father of modern philosophy—his findings in mathematics were, perhaps, equally innovative and influential. Published in 1637, La Géometrie, or The Geometry, was Descartes’s major treatise in mathematics. In it he described the principles of coordinate geometry, a system he created for studying geometry using algebraic principles and a coordinate system. Also known as “analytic” or “Cartesian” geometry, this system allowed geometric problems to be solved more easily through the use of algebra. Descartes’s creation provided a link between algebra and geometry. Cartesian geometry presented algebraic equations in the form of geometric shapes mapped onto a coordinate system. As a result of this creation, Descartes is referred to as “the father of analytical geometry.” Descartes’s mathematical writings and findings laid the groundwork for future discoveries in mathematics, such as infinitesimal calculus. The main condition necessary for the advancement of physics and astronomy that progressed during the Scientific Revolution was the advance of mathematics, which allowed the proof of abstract theories and provided a more logical method for attacking the Aristotelian system. Cantorian Set Theory Both the theory of real numbers and the idea of a function depended upon an informal notion of set. Cantor turned the very simple idea of a set into a rich theory which was to become the foundation of modern mathematics.
Scientific Revolution – progress - proof of abstract theories – use of symbols – TRIGONOMETRY - incline planes –
FRACTIONS– LOGARITHMS - trajectories of motion and cartesian co-ordinates - DIFFERENTIAL CALCULUS
TIMELINE
The Scientific Revolution and Mathematics
The main condition necessary for the advancement of physics and astronomy that progressed during the Scientific Revolution was the advance of mathematics, which allowed the proof of abstract theories and provided a more logical method for attacking the Aristotelian system.
Descartes' philosophical methodology, involving scepticism (rejection of authority) and rationalism (an assertion of self-confidence) shows the shift of mood. Though Descartes' method is deductive, he rejects (Aristotelian) logic as sterile.
This new science, most clearly illustrated in Newton's mechanics, provided the stimulus for major developments in mathematics in the 17th century and the rapid development of analysis through the 18th.
Co-ordinate geometry was introduced by Fermat and Descartes, ignoring rather than solving the foundational problems which had prevented the Greeks from taking this step (viz: the lack of any well-understood number system which could account for incommensurable ratios).
This development is important to science because it makes geometry quantitative and permits the use of algebraic methods. Geometry must be quantitative for it to be useful in science and engineering, and algebraic methods permit the more rapid development of mathematics than the less systematic (if more rigorous) methods required by the Greek axiomatic approach to geometry.
Developed independently by Newton and Leibniz, the calculus is the foundation on which a large part of the mathematics required for science is built. The development of the calculus depended upon a number system which not only includes irrational (numbers necessitated by incommensurable ratios), but even infinitesimal numbers, which sometimes behave like zero and sometimes don't.
Objections, particularly to the latter, failed to halt the onward sweep of mathematics in the service of science.
During the 18th century, the calculus was further developed and other branches of analysis were rapidly opened up. These developments lacked logical rigour and were guided by intuitive and physical insights.
Though the lack of rigour was noted, for example by Bishop Berkeley, and many attempts were made to put things right, this did not inhibit the development of the subject. It was not until well into the 19th century that rigorous foundations for this plethora of new mathematics could be discovered.
The Formalisation of Mathematics
During the period from about 1821 to 1908, which begins with the publication of Cauchy's Coursd'analysealgebraic and concludes with the publication in 1908 of Russell's Mathematical Logic as Based on The Theory of Types and Zermelo's first order axiomatisation of set theory, mathematicians restored and surpassed the standards of rigour which had been established during the period of classical greek mathematics but neglected during the mathematization of science.
The Rigourisation of Analysis
The use of infinitesimals was one of the most conspicuous practices in the development of analysis for which no justification or rationale could be offered. The rigourisation of analysis begins with the elimination of infinitesimals in favour of arguments using limits.
Frege's Logic For millenia, mathematics had been a science based on deductive logic. But no account of logic had ever been produced which was adeqate for the purposes of mathematics. By abandoning Aristotle's subject-predicate analysis of the form of sentences, Frege was able to devise a logic which would prove sufficient for the formalisation of mathematics.
Grundgesetze and Principia Mathematica:
Frege's logic was intended to provide a foundation for mathematics, and Frege showed how this could be done in his Grundgesetze der Arithmetic. This however proved to be based on an inconsistent logical system. Russell, taking the paradoxes into account, devised his Theory of Types and demonstrated with Whitehead in Principia Mathematica how this could be used to formalise mathematics. Formal mathematics was shown to be rather hard work for human beings, and did not catch on.
The Theory of Real Numbers
The rigourisation of analysis preceded and showed the need for a better account of the real number system, since the necessary proofs of results in analysis depended on properties of real numbers which were not themselves rigorously established.
Cantorian Set Theory
Both the theory of real numbers and the idea of a function depended upon an informal notion of set. Cantor turned the very simple idea of a set into a rich theory which was to become the foundation of modern mathematics.
Zermelo's Axiomatic Set Theory
After the paradoxes became conspicuous in set theory it became essential to have a consistent axiomatization of the theory if it was to be used. Zermelo's axiomatisation was the first, and with some subsequent improvements became the de-facto standard foundation for mathematics. Though Zermelo's account was informal it is a small step from there for a fully formal account of mathematics as based on first order set theory.
What changed in mathematics
LETTERS TO REPRESENT UNKNOWN QUANTITIES. During the late sixteenth century, a French lawyer, Francois Viete, was among the first to use letters to represent unknown quantities. In 1591 and after, he applied this algebraic method to geometry, laying the foundation for the invention of trigonometry.
THE INCLINE PLANES- FRACTIONS. The Fleming Simon Stevin also worked with geometry during the late sixteenth century, applying it to the physics of incline planes and the hydrostatic surface tension of water. Additionally, he introduced the decimal system of representing fractions, an advance which greatly eased the task of calculation.
INVENTION OF LOGARITHMS. However, perhaps the most important mathematical advance of the early period of the Scientific Revolution was the invention of logarithms in 1594 by John Napier of Scotland. Napier spent the next 20 years of his life developing his theory and computing an extensive table of logarithms to aid in calculation. In 1614, he published Description of the Marvelous Canon of Logarithms, which contained the fruits of these labors.
TRAJECTORIES OF MOTION AND CARTESIAN CO-ORDINATES. Johannes Kepler also did a great deal of work in geometry, which proved significant to his subsequent work in astronomy. In 1637, Rene Descartes published Geometry, in which he describes how geometry relates to motion and showed that at any moment the position of a point can be defined by its relation to surrounding planes or reference. The most well-known application of this theory is the use of a curve on a graph to represent the motion of an object, which could then be defined by a mathematical equation..
DIFFERENTIAL CALCULUS. Further progress in mathematics was made by Oxford professor John Wallis. His first work, Arithmetica Infinitum, published in 1655, set the stage for the invention and development of differential calculus. Wallis' became one of Isaac Newton's major influences. Wallis was the first mathematician to apply mathematics to the operation of the tides, and also invented the symbol used to denote infinity.
FUTHERMORE. Many mathematicians applied their knowledge to the study of optics, a field that had garnered great interest since the Middle Ages. The advances made in this field, including the development of techniques for higher resolution, led to the better construction of optical instruments, such as the telescope, which played a large part in the later work of Galileo.
Summing up
1.It was a "gradual" revolution
4. Stages of scientific development
9. The Scientific methods
13.How did Descartes justify observation?
Other Scientific Advances
During the period from about 1821 to 1908, which begins with the publication of Cauchy's Cours d'analyse algebraic and concludes with the publication in 1908 of Russell's Mathematical Logic as Based on The Theory of Types and Zermelo's first order axiomatisation of set theory, mathematicians restored and surpassed the standards of rigour which had been established during the period of classical greek mathematics but neglected during the mathematisation of scienceEurope: cultural innovations and navigation around the world.
Asia (China and Japan): 1600s: a closed or semi-closed system to maintain existent social hierarchy; where government office was worshipped by the people and technology and commerce were treated lightly by the state.
4. Stages of scientific development
Based on Hermetic theory: a divine spirit present in all the material things in the world. And the job of the natural philosopher was to capture these divine messages.
A completely secular approach to science, treating the world as consisting of a rational, knowable order.
Implied in the Hermetic theory is independent search for truth instead of accepting established truth; mathematics enabled independent thinking beyond a mere reading of the scriptures.
Mathematical conclusion can be publicly arrived at and publicly demonstrated, encouraging social consensus.
Brahe: in Denmark,. Changed established view of the stars. Royal patronage.
Kepler: Three laws of planetary motion: challenging the view of a harmonious universe through the perfect circles of the stars’ movements, and through the constant speed of these movements.
Galileo: telescope to see the moon; inertia: contradicts the traditional belief that objects are naturally at rest; the state of motion is just as natural as the state of rest.
Newton: Universal law of gravity; calculus; nature of light; mathematical rules of the three laws of motion: inertia, acceleration, action and reaction; the universe was infinite and had no center.
9. Scientific methods
To divide each difficulty into many parts and search for solution.
To arrange thoughts in order beginning with the simplest to the most difficult.
To make sure nothing is omitted.
Meticulous collection of data; and procedure of proof (from the simple to the difficult) made proof a publicly demonstrable procedure.
Clear and distinct ideas: emphasis on exactness and accuracy.
Science also emphasizes exhaustion of data, so that the results can apply to similar conditions every where in the universe.
Emphasis on procedure suggests that science had its own internal rules to follow.
Take apart a big problem and make it observable.
He bridged physics and mathematics: by reducing the whole world to mathematical representation. He also bridged observation and legitimacy by connecting observation through mathematics. Furthermore, he justified truth through mathematics and observation by connection with God.
Then you seek it in reality because it will sure be there.
In other words, explorations in the world are valid because they are backed by these "perfect ideas" of God in one’s mind. This can be used to justify empirical observations only because it made observations ‘legitimate,’ so to speak, by reconciling Catholicism and scientific observation, not because his observation was the same as our definition of observation today.
||The Rigourisation of Analysis
The use of infinitesimals was one of the more conspicuous practices in the development of analysis for which no justification or rationale could be offered. The rigourisation of analysis begins with the elimination of infinitesimals in favour of arguments using limits.
Frege's Logic
For millenia mathematics had been a science based on deductive logic. But no account of logic had ever been produced which was adeqate for the purposes of mathematics. By abandoning Aristotle's subject-predicate analysis of the form of sentences, Frege was able to devise a logic which would prove sufficient for the formalisation of mathematics.
Grundgesetze and Principia Mathematica
Frege's logic was intended to provide a foundation for mathematics, and Frege showed how this could be done in his //Grundgesetze der Arithmetic//. This however proved to be based on an inconsistent logical system. Russell, taking the paradoxes into account, devised his Theory of Types and demonstrated with Whitehead in //Principia Mathematica// how this could be used to formalise mathematics. Formal mathematics was shown to be rather hard work for human beings, and did not catch on.
The rigourisation of analysis preceded and showed the need for a better account of the real number system, since the necessary proofs of results in analysis depended on properties of real numbers which were not themselves rigorously established.
Cantorian Set Theory
Both the theory of real numbers and the idea of a function depended upon an informal notion of set. Cantor turned the very simple idea of a set into a rich theory which was to become the foundation of modern mathematics.
Zermelo's Axiomatic Set Theory
After the paradoxes became conspicuous in set theory it became essential to have a consistent axiomatisation of the theory if it was to be used. Zermelo's axiomatisation was the first, and with some subsequent improvements became the de-facto standard foundation for mathematics. Though Zermelo's account was informal it is a small step from there for a fully formal account of mathematics as based on first order set theory.
René Descartes (1596–1650) is considered one of the major figures of the seventeenth century and an important contributor to the Scientific Revolution. Descartes was a French-born philosopher and mathematician, but spent most of his adult life living in the Dutch Republic. The Dutch environment influenced the development of Descartes’s philosophical and mathematical theories. Although Descartes is primarily remembered for his writings as a philosopher—today he is often considered the father of modern philosophy—his findings in mathematics were, perhaps, equally innovative and influential. Published in 1637, La Géometrie, or The Geometry, was Descartes’s major treatise in mathematics. In it he described the principles of coordinate geometry, a system he created for studying geometry using algebraic principles and a coordinate system. Also known as “analytic” or “Cartesian” geometry, this system allowed geometric problems to be solved more easily through the use of algebra. Descartes’s creation provided a link between algebra and geometry. Cartesian geometry presented algebraic equations in the form of geometric shapes mapped onto a coordinate system. As a result of this creation, Descartes is referred to as “the father of analytical geometry.” Descartes’s mathematical writings and findings laid the groundwork for future discoveries in mathematics, such as infinitesimal calculus.
The main condition necessary for the advancement of physics and astronomy that progressed during the Scientific Revolution was the advance of mathematics, which allowed the proof of abstract theories and provided a more logical method for attacking the Aristotelian system.
Cantorian Set Theory
Both the theory of real numbers and the idea of a function depended upon an informal notion of set. Cantor turned the very simple idea of a set into a rich theory which was to become the foundation of modern mathematics.
What changed in mathematics